If $\displaystyle Z$ is a Complex no. . Then Find Min. value of $\displaystyle |2Z-1|+|3Z-2|$

Consider that $\displaystyle |2z-1|+|3z-2|=2\cdot |z-1/2|+3\cdot |z-2/3|$. Now, surely, the value of z that minimizes this expression lies on the line between $\displaystyle z_1:= 1/2$ and $\displaystyle z_2 := 2/3$ and is, therefore, a real number.
In fact, it must be $\displaystyle z = 2/3$, for if you shift a $\displaystyle z\in\mathbb{R}$ that lies between $\displaystyle z_1$ and $\displaystyle z_2$ a little closer towards $\displaystyle z_2$, without exceeding it, the first term of the sum gets larger but the second term gets smaller by a larger amount. This continues until you reach $\displaystyle z_2$; after that point on the real axis both terms increase.